Far and away the best source of postdoctoral possibilities is the AAS Job Register. New openings are posted monthly.

Later on in the process, you will waste countless hours compulsively reloading the Astrophysics Jobs Rumor Mill. This is also a good place to find a listing of fellowships to apply for, with links to the AAS job ads.

Andrew West made a script that searches the AAS Job Register and gets the job listings before they appear in the Register at the end of the month. He has a webpage of the results.

A list of fellowships and their details are:

This is only a very partial list! Check the AAS Job register regularly. Old list of fellowships you could look at which has more of a European perspective.

The majority of prestigious fellowships have application due dates in late october or early November. The NSF is earlier, around Oct 10, while many institution-specific postdoc jobs are later on, in November or December. The International NSF fellowship is even earlier, mid-September.

You'll probably find that applying to jobs is itself a full-time job for at least a month or so in the fall. Don't leave things for the last minute! Applying to some of the early fellowships (like NSF) can prepare you for the later applications.

When one hears back about a job also varies greatly. There are often interviews at the AAS Meeting in January. Most of the major fellowships (Hubble, Jansky, NSF, etc.) will announce awards during the week or two after the AAS meeting. Award notifications can come by phone or email. Though most jobs will send out rejection letters, not every job will bother, so sometimes it's annoyingly hard to tell what the status of an ongoing application is. Typically, you can ask someone about your status. They will be up front with you.

By this point - hopefully it's time for congratulations! You will typically have a few weeks to mull over your offer(s) and make a decision. February 15 is a common date for making a commitment to accept or decline a job or fellowship offer. If you are lucky enough to have multiple offers, you do have the potential to negotiate some terms. This is usually research money or responsibilities, but could be additional years at an institution. Don't be afraid to make appropriate requests.

The job market is competitive, especially for the big prize fellowships which can receive several hundred applications. Some people worry that they need to apply to 20 or 30 jobs to have any chance - but that's probably a bit alarmist. There are no really good rules of thumb, but most people seem to apply to 10-15 postdocs or so, with a large variance.

The best advice is probably this: Talk to your friends that just went through the process! Learn from their experiences. If you ask nicely, some people may even let you look at their old applications. This can be a valuable way to get a sense of what a good postdoc application looks like.

Good luck!